Moving a surface from one side of an object to the other
Select the faces for mouth, press Y to split;





Press S > X > -1 to give it a mirrored scale along its X axis. Done. (or CtrlM > X)




P.S.: I'm not so sure about the word "rig" you mentioned here, suppose you are trying to animate it, then I suggest using Blender's ShapeKey feature in this case. Demo file included.